Application of Vertex Colorings with Some Interesting Graphs

International Journal of Trend in Scientific Research and Development (IJTSRD)Volume 3 Issue 6, October 2019

@ IJTSRD | Unique Paper ID – IJTSRD29263

Application of Vertex Colorings

Department of Mathematics, Technological University (Yamethin),

ABSTRACT

Firstly, basic concepts of graph and vertex colorings are

some interesting graphs with vertex colorings are presented. A vertex

coloring of graph G is an assignment of colors to the vertices of G. And then

by using proper vertex coloring, some interesting graphs are described. By

using some applications of vertex colorings, two problems is presented

interestingly. The vertex coloring is the starting point of graph coloring. The

chromatic number for some interesting graphs and some results are

studied.

KEYWORDS: graph, vertices, edges, the chromatic numbe

I. INTRODUCTION:

Each map in the plane can be represented by a graph. To

set up this correspondence, each region of the map is

represented by a vertex. Edges connect two vertices if the

regions represented by these vertices have a common

border. Two regions that touch at only one point are not

considered adjacent. The resulting graph is c

graph of the map. By the way in which dual graphs of maps

are constructed, it is clear that any map in the plane has a

planar dual graph. The problem of coloring the regions of

the map is equivalent to the problem of coloring the

vertices of the dual graph so that no two adjacent vertices

in this graph have the same color.

II. BASIC CONCEPTS AND DEFINITIONS

A. Basic Concepts

A graph G = (V, E) consists of a finite nonempty set V,

called the set of vertices and a set E of unordered pairs of

distinct vertices, called the set of edges. Two vertices of a

graph are adjacent if they are joined by an edge. The

degree of vertex v in a graph denoted by d(v) is the

number of edges incident to v. We denoted by

�� the minimum and maximum degree, respectively of

vertices of G. If e has just one endpoint, e is called a loop. If

�� and �� are two different edges that have the same

endpoints, then we call �� and �� parallel edges. A graph is

simple if it has no loops and parallel edges

The graph ),( EVG ′′=′ is a sub graph of the graph G = (V, E)

if V ′ V⊆ and .EE ⊆′ In this case we write

but GG ≠′ we write GG ⊂′ and call G ′ is a

graph of G. The removal of a vertex �

results in that subgraph of G consisting of all vertices of G

International Journal of Trend in Scientific Research and Development (IJTSRD)2019 Available Online: www.ijtsrd.com e

29263 | Volume – 3 | Issue – 6 | September

f Vertex Colorings with Some Interesting Graphs

Ei Ei Moe

f Mathematics, Technological University (Yamethin),

Firstly, basic concepts of graph and vertex colorings are introduced. Then,

some interesting graphs with vertex colorings are presented. A vertex

coloring of graph G is an assignment of colors to the vertices of G. And then

by using proper vertex coloring, some interesting graphs are described. By

vertex colorings, two problems is presented

interestingly. The vertex coloring is the starting point of graph coloring. The

for some interesting graphs and some results are

graph, vertices, edges, the chromatic number, vertex coloring

How to cite this paper

"Application of Vertex Colorings with

Some Interesting

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International Journal of Trend in

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Journal. This is an Open Access

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by/4.0)

Each map in the plane can be represented by a graph. To

correspondence, each region of the map is

represented by a vertex. Edges connect two vertices if the

regions represented by these vertices have a common

border. Two regions that touch at only one point are not

considered adjacent. The resulting graph is called the dual

graph of the map. By the way in which dual graphs of maps

are constructed, it is clear that any map in the plane has a

planar dual graph. The problem of coloring the regions of

the map is equivalent to the problem of coloring the

the dual graph so that no two adjacent vertices

BASIC CONCEPTS AND DEFINITIONS

G = (V, E) consists of a finite nonempty set V,

and a set E of unordered pairs of

distinct vertices, called the set of edges. Two vertices of a

if they are joined by an edge. The

of vertex v in a graph denoted by d(v) is the

number of edges incident to v. We denoted by �� and

degree, respectively of

vertices of G. If e has just one endpoint, e is called a loop. If

are two different edges that have the same

parallel edges. A graph is

parallel edges.

of the graph G = (V, E)

In this case we write .GG ⊆′ GG ⊆′

is a proper sub

from a graph G

results in that subgraph of G consisting of all vertices of G

expect � and all edges not incident with

is denoted by � � �. Thus �

of G not containing� . A walk

sequence W=�� …

alternately vertices and edges,ends of � are �� . We say that W is a walk

from�� , or ��-walk

called the origin and terminus

��, ��, … , �� are its internal vertices

number of edges in it is the length

path if there are no vertex repetitions. A

to be closed if �� . A closed walk is said to be

a k- cycle is odd or even according to k is o

Two vertices u and v of G are said to be

is a uv-path in G. If there is a uv

vertices u and v of G, then G is said to be connected

otherwise G is disconnected. If

connected in G, the distance between u and v in G, denoted

by dG (u,v), is the length of a shortest uv

B. Vertex Colorings

A vertex coloring of a graph is an assignment

from its vertex set to a codomain set C whose element are

called colors.

C. Vertex k- coloring

For any positive integer k, a vertex k

coloring that uses exactly k different colors.

D. Proper Vertex Coloring

A proper vertex coloring of a graph is a vertex

such that the endpoints of each edge are assigned two

International Journal of Trend in Scientific Research and Development (IJTSRD)

e-ISSN: 2456 – 6470

6 | September - October 2019 Page 991

ith Some Interesting Graphs

Myanmar

How to cite this paper: Ei Ei Moe

"Application of Vertex Colorings with

Some Interesting

Graphs" Published

in International

Journal of Trend in

Scientific Research

and Development

(ijtsrd), ISSN: 2456-

6470, Volume-3 |

6, October 2019, pp.991-996, URL:

https://www.ijtsrd.com/papers/ijtsrd2

Copyright © 2019 by author(s) and

International Journal of Trend in

Scientific Research and Development

Journal. This is an Open Access article

distributed under

the terms of the

Creative Commons

Attribution License (CC BY 4.0)

http://creativecommons.org/licenses/

and all edges not incident with� . This subgraph

� � � is the maximal subgraph

walk in G is a finite non-null

…��, whose terms are

alternately vertices and edges, such that, for 1≤ i≤ k, the . We say that W is a walk

walk. The vertices �� are

terminus of W respectively, and internal vertices. The integer k, the

length of W. A walk is called a

if there are no vertex repetitions. A ��-walk is said

. A closed walk is said to be k-cycle;

according to k is odd or even.

Two vertices u and v of G are said to be connected if there

path in G. If there is a uv-path in G for any distinct

vertices u and v of G, then G is said to be connected,

otherwise G is disconnected. If vertices u and v are

G, the distance between u and v in G, denoted

(u,v), is the length of a shortest uv- path in G.

of a graph is an assignment �: � → �

from its vertex set to a codomain set C whose element are

vertex k-coloring is a vertex-

coloring that uses exactly k different colors.

Proper Vertex Coloring

of a graph is a vertex- coloring

such that the endpoints of each edge are assigned two

IJTSRD29263

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470

@ IJTSRD | Unique Paper ID – IJTSRD29263 | Volume – 3 | Issue – 6 | September - October 2019 Page 992

different colors. A graph is said to be vertex k-colorable if

it has a proper vertex k- coloring.

E. Chromatic Number

The vertex chromatic number of G, denoted by, is the

minimum number different colors required for a proper

vertex- coloring of G.

F. Simple Graph

A graph which has neither loops nor multiple edges. i.e,

where each edge connects two distinct vertices and no two

edges connect the same pair of vertices is called a simple

graph.

G. Example

A simple graph G and its coloring are shown in Figure

1(a) and 1(b).

Fig. 1(a) the Simple Graph G

Fig. 1(b) Coloring of the Simple Graph G

H. Example

The graph G by using proper vertex 4- coloring is shown in

figure.

Fig.2 A Proper Vertex 4-coloring of a Graph G

I. Example

We can find the chromatic number of the graph G. Coloring

of the graph G with the chromatic number 3 is shown in

figure.

Fig.3 Coloring of the Graph G with the Chromatic

Number = 3

III. SOME INTERESTING GRAPH AND COLORING OF

GRAPHS

The planar graph and by using proper

vertex coloring,

some interesting graphs are described in the following.

A. Definitions

Let G = (V, E) be a graph. A planar embedding of G is a

picture of G in the plane such that the curves or line

segments that represent edges intersect only at their

endpoints. A graph that has a plane embedding is called a

planar graph.

B. Example

The planar graph and the planar embedding of graph are

shown in figure.

Fig. 4(a) A Planar Graph Fig. 4(b) A Planar

Embedding of G

C. Definitions

Region of a planar embedding of a graph is a maximal

connected portion of the plane that remains when the

curves, line segments, and points representing the graph

are removed.

Unbounded Region in the planar embedding of a graph is

the one region that contains points arbitrary far away

from the graph.

A component of G is a maximal connected sub graph of G.

D. Induced Sub graph

For any set S of vertices of G, the induced sub graph ⟨!⟩ is

the maximal subgraph of G with vertex set S. Thus two vertices of S are adjacent in ⟨#⟩ if and only if if they are

adjacent in G.

E. Theorem

Every planar graph is 6 colorable.

Proof

We prove the theorem by induction on the number of

vertices, the result being trivial for planar graphs with

fewer than seven vertices. Suppose then that G is a planar

graph with n vertices, and that all planar graphs with n-1

vertices are 6-colorable. Without loss of generality G can

be assumed to be a simple graph, and so, by Lemma (If G is

a planar graph, then G contains a vertex whose degree is at

most 5) contains a vertex v whose degree is at most 5; if

we delete v, then the graph which remains has n-1 vertices

and is that 6-colorable. A 6-coloring for G is then obtained

by coloring v with a different color from the (at most 5)

vertices adjacent to v.

F. Bipartite Graph

A graph G = (V, E) is bipartite if V can be partitioned into

two subsets ��and ��(i.e, V = �� ∪ �� and �� ∩ �� ∅),



such that every edges � ∈ ( joins a vertex in �� and a

vertex in ��.

G. Example

The bipartite graph is shown in figure.

Fig. 5 the Bipartite Graph

H. Theorem

The graph is bipartite if and only if its chromatic number is

at most 2.

Proof

If the graph G is bipartite, with parts �� and ��, then a

coloring of G can be obtained by assigning red to each vertex in �� and blue to each vertex in ��. Since there is no

edge from any vertex in �� to any other vertex in ��, and

no edge from any vertex in �� to any other vertex in ��, no

two adjacent vertices can receive the same color.

Therefore, the chromatic number of bipartite graph is at

most 2.

Conversely, if we have a 2-coloring of a graph G, let �� be

the set of vertices that receive color 1 and let �� be the set

of vertices that receive color 2. Since adjacent vertices

must receive different colors, there is no edge between

any two vertices in the same part. Thus, the graph is

bipartite.

I. Example

A coloring of bipartite graph with two colors is displayed

in Figure.

Fig. 6 Coloring of Bipartite Graph

J. Complete Graph

The complete graph on n vertices, for n ≥ 1, which we

denote +, , is a graph with n vertices and an edge joining

every pair of distinct vertices.

K. Example

The complete graph is shown in figure.

Fig. 7 the Complete Graph

L. Example

We can find the chromatic number of +, . A coloring of +,

can be constructed using n colors by assigning a different

color to each vertex. No two vertices can be assigned the

same color, since every two vertices of this graph are

adjacent. Hence, the chromatic number of +, � �.

M. Example

A coloring of +-using five colors is shown in Figure.

Fig. 8 A Coloring of ./.

N. Complete Bipartite Graph

The complete bipartite graph +0,,, where m and n are

positive integers, is the graph whose vertex set is the

union V= �� ∪ �� of disjoint sets of cardinalities m and n,

respectively, and whose edge set is 12�|2 ∈ ��, � ∈ ��4.

Fig.9The Complete Bipartite Graph

O. Example

A coloring of +6,7 with two colors is displayed in Figure.

Fig. 10 a Coloring of .8,9

P. Example

We can find the chromatic number of complete bipartite

graphK;,< , where m and n are positive integers. The

number of colors needed may seem to depend on m and n.

However, only two colors are needed. Color the set of m

vertices with one color and the set of n vertices with a

second color. Since edges connect only a vertex from the

set of m vertices and a vertex from the set of n vertices, no

two adjacent vertices have the same color.



Q. Definition

The n- cycle, for � ≥3, denoted�, , is the graph with n

vertices, ��, ��, … , �,,and edge set

E=1��, ��6, … , �,��, , �,==��4.

R. Example

A coloring of C6 using two colors is shown in Figure 11.

Fig. 11 A Coloring of C 6

A coloring of C5 using three colors is shown in Figure 12.

Fig. 12 a coloring of C5

S. Theorem

The chromatic number of the n- cycle is given by

( )

==

=oddn

evennn ,3

,2Cχ .

Proof

Let n be even. Pick a vertex and color it red. Proceed

around the graph in a clockwise direction (using a planar

representation of the graph) coloring the second vertex

blue, the third vertex red, and so on. We observe that odd

number vertices are colored with red. Since n is even, (n-

1) is odd, Thus (n-1)st vertex is colored by red. The nth

vertex can be colored blue, since it is adjacent to the first

vertex and (n-1)st vertex. Hence, the chromatic number of

Cn is 2 when n is even.

Let n be odd. Since n is odd, (n-1) is even. For the vertices

of first to (n-1)st, we need only two colors. By first part,

we see that the color of first vertex and (n-1)st vertex are

not the same. The nth vertex is adjacent to two vertices of

different colors, namely, the first and (n-1)st. Hence, we

need third color for nth vertex. Thus, the chromatic

number of Cn is 3 when n is odd and n>1.

T. Theorem

A graph is 2-colorable if and only if it has no odd cycles.

Proof

If G has an odd cycle, then clearly it is not 2- colorable.

Suppose then that G has no odd cycle. Choose any vertex v

and partition V into two sets VandV 21 as follows,

}{ evenisvudVuV ),(:1

∈=

}{ oddisvudVuV ),(:2 ∈=

Coloring the vertices in V1 by one color and the vertices in

V2 by another color defines a 2- coloring as long as there

are no adjacent vertices within either V1 or V2. Since G has

no odd cycle, there cannot be adjacent vertices within V1

or V2.

U. Definitions

The n-spoked wheel, for � ≥3, denoted >, , is a graph

obtained from n-cycle by adding a new vertex and edges

joining it to all the vertices of the n- cycle; the new edges

are called the spokes of the wheel.

V. Example

A coloring of six-spoked wheel using three colors is shown

in Figure.

Fig. 13 A Coloring of ?@

W. Example

A coloring of five- spoked wheel using four colors is shown

in Figure.

Fig. 14 A Coloring of ?/

X. Proposition

Wheel graphs with an even number of ‘spoke’ can be 3-

colored, while wheels with an odd number of ‘spoke’

require four colors.

Y. Theorem

If G is a graph whose largest vertex- degree is then G is

(A + 1)-colorable.

Proof

The proof is by induction on the number of vertices of G.

Let G be a graph with n vertices; then if we delete any

vertex v (and the edges incident to it), the graph which

remains is a graph with n-1 vertices whose largest vertex–

degree is at mostA. By our induction hypothesis, this

graph is (A + 1)-colorable; a (A + 1)- coloring for G is then

obtained by coloring v with a different color from the (at

mostA) vertices adjacent to v.



IV. SOME APPLICATIONS OF VERTEX COLORINGS

The followings are some applications of vertex colorings.

A. Example

Suppose that Mg Aung, Mg Ba, Mg Chit, Mg Hla, Mg Kyaw,

Mg Lin and Mg Tin are planning a ball. Mg Aung, Mg Chit,

Mg Hla constitute the publicity committee. Mg Chit, Mg Hla

and Mg Lin are the refreshment committee. Mg Lin and Mg

Aung make up the facilities committee. Mg Ba, Mg Chit, Mg

Hla, and Mg Kyaw form the decorations committee. Mg

Kyaw, Mg Aung, and Mg Tin are the music committee. Mg

Kyaw, Mg Lin, Mg Chit form the cleanup committee. We

can find the number of meeting times, in order for each

committee to meet once.

We will construct a graph model with one vertex for each

committee. Each vertex is labeled with the first letter of

the name of the committee that is represents. Two vertices

are adjacent whenever the committees have at least one

member in common. For example, P is adjacent to R, since

Mg Chit is on both the publicity committee and the

refreshment committee.

Fig. 15(a) Graph Model for the Ball Committee

Problem

To solve this problem, we have to find the chromatic

number of G. Each of the four vertices P, D, M, and C is

adjacent to each of the other. Thus in any coloring of G

these four vertices must receive different colors say 1, 2, 3

and 4, respectively, as shown in Figure. Now R is adjacent

to all of these except M, so it cannot receive the colors 1, 2,

or 4. Similarly, vertex F cannot receive colors 1, 3, and 4,

nor can it receive the color that R receives. But we could

color R with color 3 and F with color 2.

Fig. 15(b) Using a Coloring to Ball Committee Problem

This completes a 4-coloring of G. Since the chromatic

number of this graph is 4, four meeting times are required.

The publicity committee meets in the first time slot, the

facilities committee and the decoration committee meets

in the second time slot, refreshment and music committee

meet in the third time slot, and the cleanup committee

meets last.

B. Example

We can find a schedule of the final exams for Math 115,

Math 116, Math 185, Math 195, CS 101, CS 102, CS 273,

and CS 473, using the fewest number of different time

slots, if there are no students taking both Math 115 and CS

473, both Math 116 and CS 473, both Math 195 and CS

101, both Math 195 and CS 102, both Math 115 and Math

116, both Math 115 and Math 185, and Math 185 and Math

195, but there are students in every other combination of

courses. This scheduling problem can be solved using a

graph model, with vertices representing courses and with

an edge between two vertices if there is a common student

in the courses they represent. Each time slot for a final

exam is represented by a different color. A scheduling of

the exams corresponds to a coloring of the associated

graph.

Fig. 16(a) The Graph Representing the Scheduling of

Final Exams

Fig. 16(b) Using a Coloring to Schedule Final Exams

Time Period Courses

I Math116, CS 473

II Math 115, Math 185

III Math 195, CS 102

IV CS 101

V CS 273

Since1Math116,Math185, CS101, CS102, CS2734, from a

complete sub graph of G, ( )Gχ ≥ 5. Thus five time slots

are needed. A coloring of the graph using five colors and

the associated schedule are shown in Fig. 16(b).

V. CONCLUSIONS

In conclusion, by using vertex colorings, we can find

meeting time slot, Schedules for exams, the Channel

Assignment Problem with the corresponding graph

models. In Graph Theory, graph coloring is a special case

of Graph labeling (Calculable problems that satisfy a

numerical condition), assignment of colors to certain



objects in a graph subject to certain constraints vertex

coloring, edge coloring and face coloring. One of them,

vertex coloring is presented. Vertex coloring is used in a

wide variety of scientific and engineering problems.

Therefore I will continue to study other interesting graphs

and graph coloring.

ACKNOWLEDGMENT

I wish to express my grateful thanks to Head and Associate

Professor Dr. Ngwe Kyar Su Khaing , Associate Professor

Dr. Soe Soe Kyaw and Lecturer Daw Khin Aye Tint, of

Department of Engineering Mathematics, Technological

University (Yamethin), for their hearty attention, advice

and encouragement. I specially thank my parent. I would

like to thank all the teaching staffs of the Department of

Engineering Mathematics, TU (Yamethin) for their care

and supports. I express my sincere thanks to all.

REFERENCES

[1] M., Skandera, Introduction to Combinatorics and

Graph Theory, Oxford University Press, Oxford

(2003).

[2] F. Harary, Graph Theory, Narosa Publishing House,

New Delhi(2000)

[3] R. Diestel, Graph Theory, Electronic Edition New

York (2000).

[4] R. J., Wilson, Introduction to Graph Theory, Longman

Graph Limited, London (1979).

[5] J. A. Bondy. U. S. R., Graph Theory with Applications,

Macmillan Press Ltd., London (1976).

[6] K. R. Parthasarathy, Basic Graph Theory, Mc Graw-

Hill, New Delhi (1994).

Education

Application of Vertex Colorings with Some Interesting Graphs